Ghimenti, Marco; Lewin, Mathieu Properties of periodic Hartree-Fock minimizers. (English) Zbl 1159.49048 Calc. Var. Partial Differ. Equ. 35, No. 1, 39-56 (2009). Summary: We study the periodic Hartree-Fock model used for the description of electrons in a crystal. The existence of a minimizer was previously shown by I. Catto et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire 18, No. 6; 687–760 (2001; Zbl 0994.35115)]. We prove in this paper that any minimizer is necessarily a projector and that it solves a certain nonlinear equation, similarly to the atomic case. In particular we show that the Fermi level is either empty or totally filled. Cited in 2 Documents MSC: 49S05 Variational principles of physics 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 81V55 Molecular physics 35Q72 Other PDE from mechanics (MSC2000) Keywords:periodic Hartree-Fock model; existence of a minimizer; projector; nonlinear equation Citations:Zbl 0994.35115 PDFBibTeX XMLCite \textit{M. Ghimenti} and \textit{M. Lewin}, Calc. Var. Partial Differ. Equ. 35, No. 1, 39--56 (2009; Zbl 1159.49048) Full Text: DOI arXiv References: [1] Bach V.: Error bound for the Hartree–Fock energy of atoms and molecules. Comm. Math. Phys. 147(3), 527–548 (1992) · Zbl 0771.46038 · doi:10.1007/BF02097241 [2] Bach V., Lieb E.H., Solovej J.P.: Generalized Hartree–Fock theory and the Hubbard model. J. Stat. Phys. 76(1–2), 3–389 (1994) · Zbl 0839.60095 · doi:10.1007/BF02188656 [3] Bach V., Lieb E.H., Loss M., Solovej J.P.: There are no unfilled shells in unrestricted Hartree–Fock theory. Phys. Rev. Lett. 72, 2981–2983 (1994) · doi:10.1103/PhysRevLett.72.2981 [4] Cancès, E., Defranceschi, M., Kutzelnigg, W., Le Bris, C., Maday, Y.: Computational quantum chemistry: a primer. In: Ciarlet, Ph., Le Bris, C. (eds.) Handbook of Numerical Analysis, vol. X. Special Volume: Computational Chemistry. Elsevier, Amsterdam (2003) · Zbl 1070.81534 [5] Cancès É., Deleurence A., Lewin M.: A new approach to the modelling of local defects in crystals: the reduced Hartree–Fock case. Comm. Math. Phys. 281, 129–177 (2008) · Zbl 1157.82042 · doi:10.1007/s00220-008-0481-x [6] Cancès E., Le Bris C.: On the convergence of SCF algorithms for the Hartree–Fock equations. M2AN Math. Model. Numer. Anal. 34(4), 749–774 (2000) · Zbl 1090.65548 · doi:10.1051/m2an:2000102 [7] Catto I., Le Bris C., Lions P.-L.: On the thermodynamic limit for Hartree–Fock type models. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(6), 687–760 (2001) · Zbl 0994.35115 · doi:10.1016/S0294-1449(00)00059-7 [8] Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995) · Zbl 0836.47009 [9] Lieb E.H.: Variational principle for Many–Fermion systems. Phys. Rev. Lett. 46, 457–459 (1981) · doi:10.1103/PhysRevLett.46.457 [10] Lieb E.H., Simon B.: The Hartree–Fock theory for Coulomb systems. Comm. Math. Phys. 53, 185–194 (1977) · doi:10.1007/BF01609845 [11] Lieb E.H., Simon B.: The Thomas–Fermi theory of atoms, molecules and solids. Adv. Math. 23, 22–116 (1977) · Zbl 0938.81568 · doi:10.1016/0001-8708(77)90108-6 [12] Lieb E.H., Solovej J.P., Yngvason J.: Asymptotics of heavy atoms in high magnetic fields. I. Lowest Landau band regions. Comm. Pure Appl. Math. 47(4), 513–591 (1994) · Zbl 0800.49041 · doi:10.1002/cpa.3160470406 [13] Lions P.-L.: Solutions of Hartree–Fock equations for Coulomb systems. Comm. Math. Phys. 109, 33–97 (1987) · Zbl 0618.35111 · doi:10.1007/BF01205672 [14] Reed M., Simon B.: Methods of Modern Mathematical Physics, vol. IV, Analysis of Operators. Academic Press, New York (1978) · Zbl 0401.47001 [15] Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1987) · Zbl 0925.00005 [16] Thomas L.E.: Time-dependent approach to scattering from impurities in a crystal. Comm. Math. Phys. 33, 335–343 (1973) · doi:10.1007/BF01646745 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.